When I was a teenager, I was given the book Men of Mathematics by E. T. Bell, and I rather enjoyed it. I know that this book has been criticized for various reasons and I might even agree with some of the criticism, but let's not digress onto that. E. T. Bell made a reasonable list of 34 of the greatest mathematicians from the ancient Greek period to the end of the 19th century. His list isn't perfect — maybe he should have included Klein or skipped Poncelet — but no such list can be perfect anyway. I think that his selection was good. He also made the careers of these mathematicians exciting and he addressed why mathematicians care about them today. It helped me learn what achievements and topics in mathematics are important.

(Just to head off discussion, the standard complaints include that the title is sexist and so is the book, that Bell was loose with biographical facts, and that he "chewed the scenery".)

After the period covered by Bell, there is a 50-70 year gap, followed by the the Fields Medals. The list of Fields Medalists has its own limitations, but it is another interesting, comparably long list of great mathematicians. Somewhat accidentally, this list also orients and motivates advanced mathematics students today.

But who are the great mathematicians in the gap itself, that is, those born between 1850 and 1900, or say between 1860 and 1910? (Or 1920 at the latest; that is what I had before.) There is a good expanded list of mathematicians with biographies at the St. Andrews site. However, it is too long to work as a sequel to Bell's book. (Not that I plan to write one; I'm just asking.) If you were to make a list of 20 to 50 great mathematicians in this period, how would you do it or who would they be? Presumably it would include Hilbert, but who else? (Poincaré was born in 1854 and is the latest born in Bell's list.)

For example, I know that there was the IBM poster "Men of Modern Mathematics" that also earned some criticism. But I don't remember who was listed, and my recollection is that the 1850-1900 period was somewhat cramped.

I decided based on the earliest responses to convert this list to community wiki. But I am not just asking people to throw out names one by one and then vote them up. If a good reference for this question already exists, or if there is some kind of science to make a list, then that would be ideal. If you would like to post a full list, great. If you would like to list one person who surely should be included and hasn't yet been mentioned, then that's also reasonable. It may be better to add years of birth and death in parentheses, for instance "Hilbert (1862-1943)".

Note: To make lists, you should either add two spaces to the end of each line, or " - " (space dash space) to the beginning of each line.

@Jose: You should post a photo of the poster here if you get the chance later.
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Kevin H. LinDec 30 '09 at 12:27

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I edited the title to include a specific time period, because "early modern" is rather ambiguous out of context. (My first assumption when seeing the title was that "early modern" was being used in the sense that it is used in the humanities, namely 1500-1800. I was happily surprised to find out that this was actually a question that interested me much more.)
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Alison MillerDec 30 '09 at 21:42

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By the way, my list -- which ends at 1910 -- nevertheless contains the first two Fields Medallists.
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Pete L. ClarkDec 30 '09 at 22:45

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I totally agree that you shouldn't entirely trust Bell's list of names. (There are various things about Bell and his book that you shouldn't trust.) He missed Klein and Lie for sure, and you could also argue for Frobenius and Christoffel. They could replace Poncelet and Sylvester, and maybe Monge.
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Greg KuperbergDec 31 '09 at 0:00

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In addition to nonBell-preFields mathematicians, I'd like to see 3 more lists: 1.Fields-caliber nonFields mathematicians of Fields medal era (R.Bott, F.Adams, V.Arnold, ...); 2. Clear-cut super attractive theorems (hm..., perhaps a separate sublist for each of the major mathematical domains); 3. Clear-cut theories. I'd like to see such lists but I don't dare to ask :-)
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Włodzimierz HolsztyńskiFeb 27 '13 at 2:14

My impression is that most contemporary analytic number theorists regard Littlewood as being stronger than Hardy. Also, I would put Ramanujan on such a list ahead of Polya. Looked at from a modern perspective, is Birkhoff as important as Lefschetz? (I would be interested to hear the argument for the affirmative.) Probably some of Castelnuovo, Enriques, or Severi should be on the list.
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EmertonDec 30 '09 at 15:42

@Everyone: By the way, I spent much longer than I expected compiling this list: several hours. I would appreciate it if, CW aside, people did not modify it according to their own tastes but simply make suggestions. If you want to make a list which is a minor variation of mine and put that in a separate answer, no problem.
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Pete L. ClarkDec 30 '09 at 22:47

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Harry, Qiaochu, etc., please take this side question elsewhere; it has reached a natural stopping point. (Many of us do care what Hardy has to say in "A Mathematician's Apology", which is not the same as agreeing with him. But still, it has become off topic.)
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Greg KuperbergDec 31 '09 at 0:52

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Indeed, its extremely weird that Kolmogorov is not in your main list.
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alexJan 13 '10 at 23:49

Nothing very scientific, I'm afraid, but I do have my favourites from this period. Of course, Hilbert, but in addition: Élie Cartan, Emmy Noether, Hermann Weyl, Paul Dirac (if one is allowed to consider him a mathematician), Kurt Gödel, André Weil, Israel Gelfand, Laurent Schwartz. And I'm sure that I'm leaving many in the inkwell, so to speak.

A more scientific approach -- I'm fully aware of its pitfalls! -- is that given our propensity for naming things after people, one could draw a list by counting attributions using the Encyclopaedia of Mathematics or the Princeton Companion as corpus.

Edit

OK, this time $\epsilon$ more scientifically. I took The Princeton Companion to Mathematics (don't ask), converted the PDF to text, analysed the word frequency and isolated the names mentioned among the answers thus far. I then fed the results to Wordle and obtained the following amusing name cloud:

I've also made available a larger PDF version of the image. Needless to say, one has to take this with a grain of salt. It's a first approximation. For example, no effort has been made to distinguish between Cartans, Artins, 'von Neumann' actually denotes the appearances of 'Neumann'. Not to mention the fact that as Greg points out in a comment below, it is heavily biased in favour of those mathematicians after whom we name things. But I take it we are aware that any such lists are bound to have an element of bias and subjectivity.

Any chance of making this image larger so the small names are more readable?
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Sam LichtensteinDec 30 '09 at 16:26

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If you click on the link that says "name cloud" you get the larger version of the image from the Wordle site. I could post a larger version here, though, if people think this is a good idea. If so, just upvote this comment and I'll do so.
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José Figueroa-O'FarrillDec 30 '09 at 16:28

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I conjecture that PCM = Princeton Companion to Mathematics. This is a useful experiment for sure, but it is biased in favor of mathematicians with things named after them.
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Greg KuperbergDec 30 '09 at 17:39

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The bias is even weirder than that. It biases towards people who have things named after them in a particular way. "Hilbert space" counts but "Noetherian ring" doesn't.
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Noah SnyderDec 31 '09 at 6:46

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@Noah: That's a good point, but in your particular example, "noetherian" (with or without capitalisation) does not appear in the PCM! I repeat: take this with a grain of salt :)
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José Figueroa-O'FarrillDec 31 '09 at 8:01

@Harry: The question of CH is the very first of Hilbert's famous list of open problems, so to portray it as outside the main stream of mathematics is absurd. Goedel's contributions to mathematical logic are staggering. His work has led to the entire fields of model theory, computability theory, proof theory and completely transformed set theory. In so many research areas, his theorems mark the point where the subjects became mature mathematical developments, and thousands of mathematicians study his ideas.
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Joel David HamkinsDec 30 '09 at 22:27

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Harry, there are innumerable example of Goedel's influence on diverse areas of "real" mathematics. The fact that it's hopeless to try to classify topological 4-manifolds follows from undecidability results that come out of Goedel's work. I could have easily chosen all sorts of undecidability results that are central to modern geometric and combinatorial group theory. Not to mention Diophantine equations...
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HJRWDec 30 '09 at 22:28

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Folks, please continue this discussion elsewhere. Maybe Harry should start a blog to develop or discuss his views. His opinion of Gödel is an unorthodox dissent, but let's not pile it on.
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Greg KuperbergDec 30 '09 at 22:34

Motivated by an e-mail from grshutt, I looked myself at the Princeton Companion to Mathematics. I discovered that the editors of the PCM answered the same question as the one that I posed here. Part VI is a collection of biographies of 95 mathematicians, plus one for Bourbaki, and the fact that the Bernoullis share a slot. They chose 37 mathematicians born between 1850 and 1920, exactly in the middle of the range that I suggested. See the table contents to see who they picked. They also wrote nice biographies, although they are much shorter and more sedate than E.T. Bell's single-author work.

It is a little embarrassing that we had this whole discussion without noticing that the PCM, with Timothy Gowers as the chief editor, had already done the same work. But it is also useful. It seems that when informed people carefully make lists of influential mathematicians of comparable length, they'll arrive at roughly the same list. Typically the intersection is about 1/2 if the lists are exactly the same length. PCM's list is longer than Bell (in the corresponding period) and mine, and shorter than Pete's, so we can look at who I (or Harrison) included but they did not, or who they included but Pete did not.

Some of the differences can be explained as a somewhat different emphasis. In listing Russell and Church, the PCM gave more credit to the philosophical end of logic. (This is also grshutt's point in in his answer posted here and in his letter.) Whether Feynman should "count" is another question with two different reasonable answers. (As I said in a comment, the argument for Feynman is that he is an architect of quantum field theory, and quantum field theory is arguably a fundamental mathematical theory, even though parts of it haven't been made rigorous.)

And I suppose that some of the differences indicate the ill-defined side of the question, or they are just random fluctuations, and aren't really worth arguing over.

Poisson (1781-1840) is way outside of our chronology, n'est-ce pas? As you say, the PCM list seems to include logic/philosophy as well. I did think about Church as an honorable mention, and I am coming around to adding Skolem as an honorable mention.
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Pete L. ClarkJan 3 '10 at 0:40

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I wrote Poisson when it should have been Poussin! One is a fish, the other a spring chicken. :-) I have to say that Poussin is an unusual choice, for instance compared to Hurwitz.
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Greg KuperbergJan 3 '10 at 2:12

Oh, you mean de la Vallee-Poussin. He was on the very first version of my list, but I deleted him because I felt he was strictly majorized by Hadamard (who also proved PNT and did a lot more besides).
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Pete L. ClarkJan 3 '10 at 2:15

I added Skolem as an honorable mention. 1887 is a tough year: Skolem and Ramanujan don't quite make the cut!
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Pete L. ClarkJan 3 '10 at 2:19

This did not fit in a comment. By popular (or at least Kevin Lin's) request, I took some photos of the IBM Mathematicians Poster on the 5th floor of the James Clerk Maxwell Building of the University of Edinburgh, just in front of the Common Room of the School of Mathematics. (My usual place of work.)

The poster only includes mathematicians who had already died at the time the poster was designed. Before the final printing other notable mathematicians who had since died were added (in name only) to the poster.

Their choice of 23 mathematicians are as follows, where I have italicised the ones who were added after the production of the poster:

Thanks a lot! There are still six names that appear on every list so far: Hilbert, Minkowski, Cartan, Noether, Weyl, von Neumann. Although Godel and Weil were still alive at that time, and at this point some kind of consensus summary would be more interesting than the dwindling intersection.
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Greg KuperbergJan 14 '10 at 0:04

That's a pretty short list! Are those the only names you would include? Or are these the best of the best?
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Jonas MeyerDec 30 '09 at 23:26

I think that this is a list focused on the development of class field theory. In other words, it highlights the key developments in algebraic number theory in the given time period.
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EmertonDec 31 '09 at 3:43

This list was not meant to be exhaustive ! I merely wanted to say that these four should be part of any list of "greats". I deliberately left out Hilbert, Ramanujan, Weil... because they were there on other lists.
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Chandan Singh DalawatDec 31 '09 at 4:25

For the sake of metahistorical accuracy (and for fun), what would a distillation of Pete's excellent list (and perhaps Greg's bookends, plus the stragglers in the 1840s) to a shortlist of 15 or 20 look like? This would better match the "density" of mathematicians in the later part of the era Bell covers, after all.

My humble proposal (not including Cantor, Kovalevskaya, and Poincaré, who are in Bell already):

That's 20, counting Elie and Henri Cartan as one a la the Bernoullis. In attempted keeping with the flavor of the original, I gave special consideration to mathematicians whose life could be "romanticized" to some degree, hence the inclusion of von Neumann, Turing and especially Feynman.

Thoughts? Keep in mind that I'm trying for "original flavor," so look at biographical considerations as well as mathematical, and play nice.

Feynman is debatable. As much as I like the guy, I'm not sure how much serious mathematical work he's done (are you thinking of the path integral?).
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Qiaochu YuanDec 31 '09 at 1:28

The path integral and the proto-quantum computing stuff. I'm not sure he belongs on mathematical merit either -- although neither am I convinced that QED should be disqualified as "serious mathematical work" just because it happens to describe our universe -- but I'm going for an E.T. Bell vibe, and he was a really interesting guy.
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Harrison BrownDec 31 '09 at 1:44

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The case for Feynman is that he is an architect of quantum field theory. (Not just QED by any means.) If we are to include Lie as an architect of Lie groups and Weil as an architect of arithmetic geometry, then we should also think about the position of quantum field theory. Much of QFT is non-rigorous mathematics, but maybe that's our fault as mathematicians, because arguably it is profound mathematics.
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Greg KuperbergDec 31 '09 at 2:09

To extend Pete Clark's very nice work, I imitated his methods for the decades before and after the period that he covered. It is exhausting to keep making one decision after another, and I didn't bother separating people into main list and honorable mention. Maybe Pete or someone else can help with the latter — feel free to edit this answer!

It's clearly time to stop around 1920, not just because of the Fields Medal, but also because more and more mathematicians born after then are still alive. Also, I understand that Feynman is a debatable choice as a mathematician, but I think that he should at least be considered.

I added Burnside, who hopefully won't be too controversial, to this list, and Garrett Birkhoff, who I fear will be. Note that Frobenius misses the 1950 cutoff by 2 + epsilon months.
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Harrison BrownDec 30 '09 at 23:32

By which I mean the 1850 cutoff, of course...
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Harrison BrownDec 30 '09 at 23:33

Pete, as much as I want you to retain final control of your long answer, I'd like to let you edit these bookends. Maybe the list of 20 that I posted below will be the one that stays with me.
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Greg KuperbergDec 31 '09 at 1:37

I'm not sure that Garrett Birkhoff merits inclusion (and thus I confirm the prediction of controversy). On the other hand, I agree with the inclusion of Burnside.
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EmertonDec 31 '09 at 3:49

I was going to request an answer like Harrison's, then I was going to post an answer like Harrison's, but Harrison scooped me. So by a happy accident, we have two independent lists of 20 people. My list has a slightly narrower time frame. I agree about Lie and Klein, but arguably they should have been in E. T. Bell's book rather than in the list compiled here. I concede that I also see Harrison's argument to include them here (if you also don't skip Cantor, Kovalevskaya, and Poincaré). Also, I respect Harrison's point about including people whose lives can be romanticized, and I even agree that a biography should try to romanticize the lives of mathematicians, but I think that the math should come first in deciding who to write about. (Besides, Erdős, Feynman, and Turing have already provided enough human interest for biographers.)

Anyway, here is my list. (I cheated in just one spot; I replaced one of mine by Kolmogorov.)

Are there other things to say about the symmetric difference, (Harrison's list) ∆ (my list)? We intersect in 11 people; should our lists be made to converge? Are there fundamental mathematicians who we both missed?

For what it's worth: Picard was almost on my list but was a casualty of the decision to emphasize "romanticizable" mathematicians, and Artin and either Peano or Hurwitz would probably have taken the places of Lie and Klein. Apart from that, I think I might be somewhat biased toward analytic number theorists (Hadamard was another last-minute cut), probably because books about the Riemann hypothesis were my E.T. Bell. That would explain Hardy and Littlewood.
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Harrison BrownDec 31 '09 at 1:34

I fought against my own bias, then I caved and decided that I should include Minkowski even if I am biased in his favor. As for Picard, look at what the St. Andrews site says: Hated math at first, but studied it to avoid punishment! Three children killed in World War I! Educated 10,000 French engineers! Compelling human interest comes in many guises.
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Greg KuperbergDec 31 '09 at 1:47

out of which only Chern was born before 1920.
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Gjergji ZaimiDec 30 '09 at 8:39

Um, no, the EARLY modern period, before 1920 at the very latest. Even Erdos is at the tail end of the question, since he was just a few years older than several of the Fields Medalists.
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Greg KuperbergDec 30 '09 at 8:40